The Optimization of Base Bleed Grain Parameters for ... · An optimization technique will be used to study the effect of different parameters of the grain including outer radius,

Paper: ASAT-16-171-AE

16th

International Conference on

AEROSPACE SCIENCES & AVIATION TECHNOLOGY,

ASAT - 16 – May 26 - 28, 2015, E-Mail: [email protected]

Military Technical College, Kobry Elkobbah, Cairo, Egypt

Tel : +(202) 24025292 – 24036138, Fax: +(202) 22621908

The Optimization of Base Bleed Grain Parameters for Maximum

Ballistic Performance

H.A. Abou-Elela*, Ossama R*, A.Z. Ibrahim*, O.K. Mahmoud* and O.E. Abdel-Hamid*

* Egyptian Armed Forces

ABSTRACT

Solid propellant Base bleed unit is one of the effective methods to increase the range of

artillery projectiles. As far as we search, no published study focuses on the optimal

dimensions of the base bleed grain. However, few studies focus on the effect of base bleed

grain parameters on its ballistic performance in which each parameter was studied separately.

The present optimization study is performed on base bleed grain which performed as

longitudinally slotted tubular cylinder. Different case studies have been introduced according

to the number of design variables which are: grain outer, inner radius, length, burn rate, base

bleed grain unit orifice diameter. Moreover, the study is extended to demonstrate the effect of

these parameters on the innovative multi-burn rate base bleed grain. In this new idea, the grain

is splitted into two horizontal parts one with higher burn rate than the other part. The idea is to

have a grain that provides high mass flow rate in the first seconds of projectile flight, while

keeping long bleeding time.

The optimization constrains are the upper and lower limits of each design variable. An

analytical model has been developed in C++ environment to accurately evaluate the range of

the projectile. This model is then utilized in combination with design of experiment (DOE)

and the response surface method (RSM) to develop a smooth response function which can be

effectively used in the design optimization formulation as the objective function. The

objective of the optimization is to find the design variables which contribute the maximum

range.


The results show the importance of applying optimization and provided the optimized values

of the studied parameters at each case. Also it shows that the application of the new idea of

multi-burn rate base bleed increases the range in all cases with percentage up to 12 % with

respect to the range increase for the original base bleed grain.

KEY WORDS Aerodynamics, base bleed, injection parameter, Burn rate, Optimization, ANSYS, RSM.

Introduction

Long range is the significant requirement for the developers of new ammunition. Base drag

reduction is one of the main concerns to increase the range since it represents more than 50%

of the total drag affecting the projectile at transonic and supersonic speeds [1]. Base bleeding

is an effective means to cope base drag via injecting hot burnt gases behind the projectile

base. These gases raise the wake region pressure causing base drag reduction.

The effect of base bleeding is characterized by dimensionless injection parameter “I” [2-5].

Experimental work done by (Davenas) [6] showed that the optimum injection parameter, Iop

for base bleed projectile is approximately 0.005 and is calculated according to the following

equation [6, 7]:

(1)

where is the mass flow rate of burnt gases through the base bleed nozzle and is the

upstream mass flow rate of air past the projectile base which can be determined using the

following equation [7]:

, (2)

where

, are the free stream density and velocity, respectively, Ab is the area of the boat-

tail base. The mass flow rate ( ) is a function of the burning rate of base bleed grain

composition and the exposed instantaneous grain surface (Abb).

, (3)

where bb and U are the density and burning rate of the base bleed grain composition,

respectively. The grain burning rate can be calculated according to the following equation [7,

8]:


, (4)

where k is the spin rate factor, Uo is the grain burning rate at atmospheric pressure, Pch is the

pressure of the base bleed unit chamber and is the pressure exponent.

Based on the value of mass flow rate of the bleeded gases [5] as shown in Figure (1), there are

three regimes of base flow according to the values of injection parameter with respect to the

optimum value of it. As the injection parameter increases up to optimum value, the strength of

Primary Recirculating Region (PRR) which lies behind the projectile base decreases and the

recompression shock is weakened and consequently increases base pressure reaching the

maximum value at optimum injection parameter. If the injection parameter exceeds its

optimum value, the projectile base pressure decreases until the injected gases have enough

momentum to penetrate through the primary recirculation region then the pressure may

decrease to a value less than its counterpart without base bleed unit as demonstrated in the

experimental work discussed in Ref. 3 and the analytical work in Ref 9.

Fig.1 Base pressure versus injected mass flow rate [5]

S. Jaramaz et. al [7] introduced mathematical model that calculates the ballistic performance

of base bleed grain. They used it to study the effect of some of the grain parameters on range

such as number of segments, burning rate, inner diameter, grain length and base bleed unit

orifice. The study was performed in case of 3 and 6 segment grain. They outlined that both the

grain length and burn rate have the most important parameters. However they performed the

study for each parameter individually without subjecting the results to optimization technique

and they excluded grain outer diameter. Also they did not introduce explanations for these

outcomes.


H. Ali et. al. [9] used a validated mathematical model [8] that has been established based on

the model introduced by S. Jaramaz et. al [7] to study the effect of different parameters of

projectile 155mm K307 with base bleed. The base bleed grain is slotted tubular in shape with

2 slots. They included the above mentioned base bleed grain parameters besides the study of

the effect of grain outer diameter. They redressed the shortage S. Jaramaz et. al [7] by

introducing an explanation for each outcome in conjunction with the time history change of

different burnt area of base bleed grain (cylindrical – slots – total), injection parameter and the

ballistic parameters of the projectile during base bleed burning. The deeper understand of the

results lead to introduce two new techniques to control mass flow rate that ensure the

generation of optimum injection parameter; deformable exit diameter, dual-burn base bleed

grain. However the results, the study suffered like the previous by not including optimization

of the different parameters.

Fig. 2. Base bleed grain different surfaces [9].

Design optimization based on finite element model or computational fluid dynamics is

computationally very expensive and may not render accurate optimum results due to the noisy

nature of the finite element response. Here, design of experiment (DOE) and the response

surface method (RSM) have been utilized to develop a smooth response function (Range of

the Shell). DOE has been used to find the best possible combinations of the assigned design

variables which cover all the design space. Then using the finite element model, the

maximum range has been calculated for each combination to complete the DOE matrix. Later,

the Response Surface Method (RSM) has been used to illustrate response surface with the

different design variables using a fully quadratic polynomial equation. Finally, the developed

objective function has been minimized using first Genetic Algorithm to find the near global

optimum solution.

The present study is applied to a base bleed projectile model K307 launched with muzzle

velocity 910 m/s at angle of fire 51.2 which corresponds to the maximum range. Base bleed

rmx

Li

rs

ac as1

rm

x ri ωi

r ω


grain consists of two identical solid propellant grains as shown in Fig. 2 introduced by H. Ali

et. al. [8]. This configuration introduces burnt area of an inner cylinder and four flat slots

surfaces. An optimization technique will be used to study the effect of different parameters of

the grain including outer radius, Rmax, inner radius, Rin, grain length, L, base bleed unit exit

diameter, Dexit, burn rate, BR1, and number of slots Ns.

For the new multi-burn rates grain, two parameters will be added; the higher burn rate, BR2

and grain length ratio L% which describes the length of the part with the higher burn rate with

respect to grain length. The threshold of the optimization is to achieve maximum range that

will be calculated via the already validated C++ model [8]. Different cases of parameters

combination will be studied and the outputs will be represented.

Base bleed design variables

In this study, a semi-analytical model compiled with C++ [8] is fed by the values of base

bleed grain variables. The number of variables changes according to each studying case. For

each variable, its value changes within a suggested range. Table 1 shows the different cases

and the range of each variable. The output of the model which is maximum range will be fed

into optimizer mechanism.

The C++ model consists of different subroutines such as; 1) A 2d point mass trajectory in

which the inputs are the total drag coefficient at different Mach numbers and the

corresponding base drag coefficients. CFD was used to split base drag from the total drag. 2)

Base bleed grain instantaneous burnt surface area. The surface area is splitted into cylindrical

and flat slots areas. The burnet area is consumed according to the burn rate see eq. (3). 3)

Base drag reduction subroutine; which calculates the drag reduction during base bleed burn

time as function of injection parameter, as shown in Ref.[8]

The validation of the model was performed by comparison of the predicted maximum range,

drag coefficient versus Mach number and projectile altitude time history with the

experimental data of the projectile 155mm K307 with base bleed with muzzle velocity = 910

m/s and angle of fire 51.5 degree that corresponds to maximum range.

The results of the model [9] and refs. [11, 12] revel the conditions of getting the maximum

range. These conditions are: 1) the base bleed grain should generate injection parameter equal

or close to the optimum value (I = 0.005) on the trajectory especially at the first 10 seconds of

projectile flight with high supersonic velocity. Consequently, the drag reduction is a

maximum. Generally, in the first seconds injection parameter is smaller than optimum value


under the effect of the high values of the free stream conditions as shown in equ. (2) The burn

time of base bleed grain should be equal to half or more of the flight time of the projectile.

Different techniques were implemented to control mass flow rate to achieve these conditions

such as the change of burn rate, change the base bleed grain configuration and increase the

number of slots. The increase of number of slots leads to increase in the burnt area and

consequently generates higher injection parameter in the required zone.

In the new introduced technique, the multi-burn rates grain consists of two horizontal parts,

the first is manufactured with original burn rate, BR1 which varies from 0.9 up to 1.3 mm/s.

the second part is manufactured with higher burn rate, BR2 that varies from 1.7 up to 2.5

mm/s and its length is determined by grain length ratio, L%. This ratio varies from 0.1 up to

0.5 with respect to the total length of the grain length, L. The burn rate of BR2 provides

higher mass flow rate which is required in the first 10 seconds while the BR1 ensures long

burn time up to half of the projectile time of flight.

Case studies for the original base bleed grain

Different cases were studied; the first case is to study the effect of the change of Ns while

keeping the other parameters constant and equal to the parameters of the original grain of the

projectile K307. In the presiding cases the variable studied parameters are applied gradually

up to case #4 in which it includes all the parameters; outer grain radius, inner radius, exit

diameter, burn rate,. The studies were performed in case of different numbers of slots varies

from 2 up to 4. Table 1 shows the cases, the studied variables and the range of each case.

It has been shown in Table 1 that the design variables ranges from (1) variable in case #2 up

to (5) in the last case.


Table 1. Studied cases of the original base bleed grain, upper and lower constrains of each

design variable

Optimization Case

study designation

Base bleed studying variables upper and lower limit Number

of slots Outer

radius

Inner

radius Length

Exit

diameter

Burn

rate

Rmax,

[mm]

Rin,

[mm]

Li,

[mm]

Dexit,

[mm]

BR1,

[mm/s] Ns

Case # 1, PBB 1 60 20 94 44 1.1

2-3-4 Case # 2, PBB 2 60 20 94 44 0.9- 1.5

Case # 3, PBB 3 60 18-27 94 40 - 56 0.9- 1.3

Case # 4, PBB 4 40 – 70 18-27 75 -135 40 - 56 0.9- 1.3

Case studies for the multi-burn rates base bleed grain

In case of multi-burn rates base bleed grain, the previously studied cases are also studied

including the effect of L% and BR2. In the presiding cases the variable studied parameters are

applied gradually while keeping the other parameters constant and equal to the parameters of

the original grain of the projectile K307 up to case #4 in which it includes all the parameters.

The studies were performed in case of different numbers of slots varies from 2 up to 4. Table

2 shows the cases, the studied variables and the range of each one.

Table 2. Optimization cases of the multi-burn base bleed grain, upper and lower constrains of each

design variable

Case study

designation

Base bleed studying variables upper and lower limit

Number

of slots

Original variables New variables

Outer

radius

Inner

radius Length

Exit

diamete

r

Burn

rate

Length

ratio

Burn

rate

Rmax,

[mm]

Rin,

[mm]

L,

[mm]

Dexit,

[mm]

BR1,

[mm/s

]

L%, BR2,

[mm/s] Ns

case # 1, MBB 1 60 20 94 44 1.1

0.1 –

0.5 1.7-2.5 2-3-4

Case # 2, MBB 2 60 20 94 44 0.9-

1.3

Case # 3 , MBB

3 60 18-27 94 40 - 56

0.9-

1.3

Case # 4 , MBB

4

40 –

70 18-27

75 -

135 40 - 56

0.9-

1.3


It has been shown in Table 2 that the design variables are the first case is (2) and in the last

case are (7).

Design Optimization Formulation

In the optimization problem, design requirements or objectives are identified as maximization

of range of the projectile K307. The total design variables have been considered to be Rmax,

Rin, L, Dexit, BR1, L% and BR2. Also constraints in the form of upper and lower limits have

introduced as shown in tables 1, 2.

Derivation of Objective Functions

The C++ model [8] was used to evaluate accurately the range of the shell and consequently

the established objective functions for each combination of the different design variable. As

mentioned before, the established objective is maximization of the range of the shell within

the boundaries of the prementioned design variables. To formulate the design optimization

problems, one may combine the C++ model with optimization algorithms, however this

would be computationally very expensive due to the iterative nature of the optimization

problem in which at each iteration the objective functions may be evaluated (running the C++

model) several times. Besides, the optimal results may not be accurate due to the possible

noisy nature of the output response and also difficulty to establish the derivative of the

objective functions required for higher order optimization algorithms.

Considering the above, in this study design of experiment (DOE) and response surface

method (RSM) combined with the C++ model are effectively used to derive desired objective

functions which will be directly related to the design variables for each combination. DOE has

been used to identify the best location of design variables (design points) to accurately map

the given design space for each combination of different design variables. Once the DOE

matrix has been established, the maximum values of the response (Range of the shell) have

been calculated using the C++ model for each row (design point) in the DOE matrix. Then,

RSM based on fully quadratic response function has been used to relate the variations of

range of the shell with respect to different design variables for each combination. Finally,

these response functions have been effectively utilized as objective functions in the design

optimization problems. In the following, brief discussion regarding DOE and RSM to derive

DOE matrices and response surfaces are presented, respectively.


Design of Experiment matrices

Design of Experiments, DOE, is a tactic to develop an experimentation strategy that

maximizes learning using a minimum of resources. In many applications, the scientist is

constrained by resources and time, to investigate the numerous factors that affect these

complex processes using trial and error methods. Instead, (DOE) is an influential tool that

permits for multiple input factors to be manipulated determining their effect on a desired

output (response). By manipulating multiple inputs at the same time, DOE can recognize

important interactions that may be missed when experimenting with one factor at a time. All

possible combinations can be investigated (full factorial) or only a portion of the possible

combinations (fractional factorial) [13].

In this study, central composite design with enhanced template technique has been used which

gives 2*(2*K+2(k-f)

) factorials plus the central point with the total of 29 different

combinations where, k is the number of the design variables and f is the fractional number

which depends on the value of k (f= 0.0 for k < 5 and

f = 1 for k = 6, 7) [13]. Table 3 shows the DOE Matrix for the original base bleed case study

#3, PBB3 after finding the corresponding range of the shell using the code C++. In which, it

has (3) variables as described in Table 1.Optimization Techniques.

The developed approximate response surface functions can now be effectively used in the

design optimization problems which aim at finding optimum design variables to satisfy the

Maximum Range of the shell.

In this work, Genetic Algorithm (GA) [18] has been employed to accurately capture the

optimal configurations for each combination. It is important to mention that the GA has been

conducted using the ANSYS 14.5 optimization Design Exploration toolbox.


Table 1. DOE Matrix for the third combination, PBB 3 in case of Ns = 2

DVs

Rin, [mm] Dexit, [mm] BR1, mm/s Range,

Xop [m] Combinations

1 22.5 48 1.1 40462.7 2 18 48 1.1 40179.3

3 20.25 48 1.1 40326

4 27 48 1.1 40581.7

5 24.75 48 1.1 40569.9

6 22.5 48 0.9 39717.6

7 22.5 48 1 40151.3

8 22.5 48 1.3 40731.9

9 22.5 48 1.2 40660.9

10 22.5 40 1.1 40080.6

11 22.5 44 1.1 40436.4

12 22.5 56 1.1 40459.3

13 22.5 52 1.1 40462.5

14 18 40 0.9 39438.7

15 20.25 44 1 40014.3

16 27 40 0.9 40084.7

17 24.75 44 1 40302.3

18 18 40 1.3 39493

19 20.25 44 1.2 40489

20 27 40 1.3 39122.5

21 24.75 44 1.2 40579.6

22 18 56 0.9 39220.6

23 20.25 52 1 39977.6

24 27 56 0.9 39985.4

25 24.75 52 1 40278.3

26 18 56 1.3 40650.5

27 20.25 52 1.2 40575.1

28 27 56 1.3 40553.8

29 24.75 52 1.2 40722.9

RESULTS and DISCUSSIONS

In the following, the results of optimization technique for each case study mentioned in

Tables (1, 2) have been introduced. It includes; the optimized design variables with its

corresponding predicted maximum range, Xop for three candidate points. The C++ code [8]

will be fed by these values and the output range, Xanlt will be compared with Xop. The relative

difference between the two values of range for the same design variables at each candidate

point will be calculated as shown in equation (7):

(7)


Eqn. (8) has been used to calculate difference in percentage of range increase between the

new idea MBB and the original base bleed projectile P-BB:

(8)

where, and

are the maximum range when feeding the C++ code with the

corresponding candidate points in case of the new multi-burn base bleed,

MBB and the original base bleed, PBB, respectively. is the analytical predicted range of

the projectile with inert base bleed [8].

Also, the results will include samples of optimization outputs such as local sensitivity and

response surface for selected cases and outputs from C++ code. The cases of the original

base bleed grain, PBB

Table 4 shows the optimized design variables at each candidate point and the corresponding

range , and the relative difference in range.

In the first case, all the parameters are constant and equal to the values of K307 projectile

grain and just showing the change of range with Ns. In the second and third cases, it has been

observed that the optimized values of BR1 tend to be close to the upper limit as BR1 is the

only source of mass flow rate. however, in the last case the optimized values of BR1 tend to

be close to the lower limits because in this case the design variables includes Rmax and L

which can be considered as the main source of mass flow rate. For cases (2, 3 and 4) it

appears that the lower value of Rin is optimal value. Meanwhile, higher values of Dexit are

required for longer range to allow the high mass flow rate to flow with reasonable injection

parameter [9, 12]. The maximum difference between , and is found in the case PBB

4 in which the number of design variables is the maximum.


Table 4. Studied design variables at each candidate point, the corresponding range Xop, Xanlt and the

relative difference in range.

No. Configuration

Number

of slots,

NS

Optimized Design Variable Predicted Range Abs.

Relative

Differ. in

Range[%]

Rmax,

[mm]

Rin,

[mm]

L,

[mm]

Dexit,

[mm]

BR1,

[mm/s]

,

[m]

,

[m]

1 Case # 1,

PBB1

2 - - - - 1.1 - 40516 - 3 - - - - 1.1 - 41236 -

4 - - - - 1.1 - 41717 -

2 Case # 2,

PBB2

2

- - - - 1.26 40702 40703 0.00%

- - - - 1.23 40693 40696 0.01%

- - - - 1.34 40665 40661 0.01%

3

- - - - 1.26 41436 41440 0.01%

- - - - 1.23 41426 41434 0.02%

- - - - 1.33 41396 41389 0.02%

4

- - - - 1.23 41862 41872 0.02%

- - - - 1.20 41853 41866 0.03%

- - - - 1.30 41826 41826 0.00%

3 Case # 3,

PBB3

2

- 18.1 - 48.9 1.25 40846 40689 0.388%

- 26.2 - 49.6 1.23 40794 40651 0.352%

- 18.2 - 53.6 1.26 40791 40728 0.154%

3

- 18.1 - 56 1.30 41663 41613 0.121%

- 19.3 - 54.7 1.28 41599 41561 0.093%

- 18.7 - 50.5 1.28 41571 41585 0.033%

4

- 18 - 53.8 1.28 42144 42100 0.105%

- 18.1 - 48.9 1.25 42126 42084 0.100%

- 18.1 - 44.2 1.28 42117 42131 0.034%

4 Case # 4,

PBB4

2

52.7 18 129.9 56 0.90 42927 40466 6.08%

59.9 18.1 99.33 54.1 0.92 41935 39944 4.98%

56.1 18.4 114.4 54.7 1.08 41777 40738 2.55%

3

58.3 18 114.4 55.4 0.90 43324 41223 5.10%

63.1 18.2 115.5 48.8 0.91 42959 41714 2.98%

49.9 18.1 123.6 52.5 0.94 42405 40771 4.01%

4

63.0 18 86.23 55.1 1.30 43231 42123 2.63%

59.9 18.1 99.33 54.1 0.92 43040 41350 4.09%

63.1 18.2 115.5 48.8 0.91 42948 42437 1.20%

To increase the accuracy in the case PBB 4, the upper and the lower limits of the variables

have been reduced and then being optimized again. Table 5 shows the new reduced upper and

lower limits for the design variables which in such case


Table 5. Reduced upper and lower limits of the design variables for the case PBB 4 R

Optimization Case

study designation

Base bleed studying variables upper and lower limit Number

of slots Outer

radius

Inner

radius Length

Exit

diameter

Burn

rate

Rmax,

[mm]

Rin,

[mm]

Li,

[mm]

Dexit,

[mm]

BR1,

[mm/s] Ns

Case # 4, PBB 4 R 52.5 – 70 18-27 90 -125 40 - 56 0.9- 1.3 2-3-4

Table 6 shows the results after performing the above mentioned limits. It can be seen that the

error have has been reduced significantly and the values of Xanlt has been used in the next

comparisons.

Figure 3 shows the comparison between the range of the projectile in case of ordinary base

bleed grain, PBB 1 and the range in case of PBB 2 for different values of Ns and the percent

of range increase, respectively.

For all values of Ns, the range is longer in case of PBB 2 in which BR1 is the only optimized

variable and its value is higher than the original burn rate of the projectile grain (see Table 4).

Table 6. Studied design variables at each candidate point, the corresponding range Xop, Xanlt and the relative

difference in range for PBB 4 R.

No. Configuration

Number

of slots,

NS

Optimized Design Variable Predicted

Range, X [m]

Abs.

Relative

Differ.

in

Range

[%]

Rmax,

[mm] Rin, [mm]

L,

[mm]

Dexit,

[mm]

BR1,

[mm/s]

Xop,

[m]

Xanlt,

[m]

1

Original base

bleed # 4

reduced, PBB

4 R

2

58.6 18 132 56 0.90 42317 40855 3.58

63.2 20.38 127 55.5 0.91 41942 41006 2.28

58.3 18.05 126 52.5 0.94 41831 40794 2.54

3

58.5 18 131 56 0.90 43271 41756 3.63

63.2 20.38 127 55.5 0.91 42778 41905 2.08

65 21.5 121 56 0.94 42549 41904 1.52

4

58.3 18 131 56 0.90 43802 42446 3.19

58.3 18.09 126 52.5 0.94 43407 42430 -2.30%

63.2 20.4 127 55.5 0.91 43248 42494 -1.77%

Figure 4 shows the predicted change of injection parameter with burn time in the above

mentioned cases. For all values of Ns, Burn time of optimized base bleed grain, PBB 2 is less

than the corresponding time of PBB 1. This is related to the higher value of BR1 in case of

PBB 2 (see Table 4) where as BR1 in the case PBB 1 is 1.1 mm/s. Another result of the


higher BR1 is the higher injection parameter in most of burn time. Despite the decrease in

burn time, the high injection parameter in the first seconds of burn time compensates the

negative effect of burn time reduction and so the range increases.

Figure 5 shows the change of projectile Mach number versus time of flight for

PBB 2 with 2 and 4 slots. It is clear the benefit of the increase of injection parameter in the

first seconds and the decrease of it in the remaining burn time which gets the values of

injection parameter closer to the optimum value as shown in Fig.4.

Figure 6 shows the response surface of range with BR1 in case of PBB 2 for

Ns = 2, 3 and 4 respectively. Generally, the range increases with the increase of BR1 up to an

optimum value and beyond this value the range decreases. The optimum values of BR1

decreases with the increase of Ns as it can be seen in Table 3. The increase of Ns is

considered one of the main mechanisms to increases mass flow rate and consequently the

optimized value of BR1 is reduced with the increase of Ns for the same exit diameter, Dexit

[9].

Figure 7 shows the response surface of range change with grain Rmax for different grain L in

the case PBB 4R for Ns = 2 and 4. The high effect of Rmax change for all values of L on range

is obvious. The trend of range change with

Rmax in figures (a) and (b) is the same; the range increases with the increase of Rmax till an

optimum value which depends on the value of L and Ns. The optimum Rmax is higher in case

of low value of L. also for the same L, the optimum value of Rmax is higher in case Ns = 2

than in case of Ns = 4.

Fig. 3. The predicted range in case PBB 1 compared with the range of the case of PBB 2

for different NS.


Fig.4. The predicted change of injection parameter with burn time in case of

PBB 1 compared with the corresponding injection parameter of the case of PBB 2 for

different NS.

Fig. 5 Change of projectile Mach number versus time of flight for PBB 2 with

Ns = 2 and 4.


Fig.6 The response surface of range change with BR1 in the case PBB 2 for different NS.

(a) (b)

Fig. 7 The response surface of range change with grain Rmax for different grain L in the case

PBB 4 when Ns = 2 (a) and Ns = 4 (b).


The cases of the multi-burn base bleed grain

Table 7 shows the optimized design variables at each candidate point and the corresponding

range , and the relative difference in range. For all cases it appears that the lower

value of Rin is preferred. Meanwhile, higher values of Dexit are required for longer range to

allow the high mass flow rate to flow with reasonable injection parameter. The maximum

error was found in the case MBB3 where the number of design variables is the maximum.

However the error level with respect to PBB 4 R errors was satisfactory. Table 8 shows the

comparison between the maximum values of Xanlt for each case for PBB and its corresponding

value for MBB. As shown for all cases, the range increases when using multi-burn base bleed

grain (MBB) compared with the corresponding range of original base bleed grain (PBB).

The percentage of range increase is maximum in case MBB4 when Ns = 2 and equals to 15 %.

With the increase of the number of slots, the gain in the range by applying the multi burn

grain decreases as seen for all cases. This may be justified by that, two ways to re-configure

the grain to increase the range (increase the Ns and multi-burn base bleed grain) are applied in

the same time leads to increase in the injection parameter more than required and a reduction

in burn time and hence reduces the efficiency of base bleed.

Table 9 shows the effect of Ns for both configurations (PBB – MBB); it is clear that the range

increase with the increase of number of slots for both cases. However the percentage is less in

case of MBB because of the already achieved gain in range by the use of MBB.

Table 10 shows that the range increases for cases (2, 3 and 4) with respect to

case # 1 for both PBB and MBB in case of different Ns. With the incorporation of more

design variables in the study, the percentage of range increase gets higher. The highest

percentage occurs in case #4 in which all the parameters of base bleed grain are included in

optimization formulation.


Table 7. Optimized design variables at each candidate point, the corresponding range Xop, Xanlt and the

relative difference in range.

No. Conf.

No. of

slots,

NS

Optimized Design Variable Predicted Range, X

[m]

Abs.

Relative

Differ. in

Range [%] Rmax,

[mm

]

Rin,

[mm

]

L,

[mm

]

Dexit,

[mm

]

BR1,

[mm/s] L%,

BR2,

[mm/s

]

,

[m]

,

[m]

1

Case #

1,

MBB1

2

- - - - - 0.34 2

41616

.3

41586 0.1%

- - - - - 0.35 1.94

41614

.65

41571 0.1%

- - - - - 0.337

2.13

41609

.6

41606 0.01%

3

- - - - - 0.347 1.7 41888

.81

41781 0.3%

- - - - - 0.34 1.78 41872

.95

41801 0.2%

- - - - - 0.32 1.87 41857

.23

41817 0.1%

4

- - - - - 0.346 1.7 41947

.03

41908 0.09%

- - - - - 0.315 1.78 41930

.36

41914 0.04%

- - - - - 0.103 2.44 41919

.97

41881 0.09%

2

Case #

2,

MBB2

2

- - - - 1.18 0.31 2.13 41638 41657 0.05%

- - - - 1.19 0.32 2.05 41635 41632 0.01%

- - - - 1.16 0.33 1.98 41629 41614 0.04%

3

- - - - 1.17 0.34 1.70 41935 41813 0.29%

- - - - 1.18 0.31 1.82 41911 41845 0.16%

- - - - 1.14 0.31 1.92 41886 41857 0.07%

4

- - - - 1.14 0.33 1.70 41997 41930 0.16%

- - - - 1.12 0.28 1.78 41973 41940 0.08%

- - - - 1.19 0.23 1.85 41964 41960 0.01%

3

Case #

3,

MBB3

2

- 19.8 - 46.6 1.17 0.50 2.13 41675 41433 0.580%

- 18.5 - 55.6 1.17 0.35 2.16 41675 41687 0.029%

- 19.0 - 45.6 1.18 0.45 2.18 41671 41580 0.219%

3

- 19.0 - 51.3 1.26 0.20 2.50 42066 42053 0.03%

- 18.5 - 42.2 1.14 0.28 1.83 42034 41953 0.19%

- 18.1 - 48.9 1.25 0.28 1.72 42033 41942 0.22%

4

- 18.0 - 54.3 1.07 0.50 1.70 42274 42041 0.55%

- 18.2 - 43 1.16 0.24 1.75 42160 42154 0.02%

- 18.4 - 48.5 1.12 0.49 1.70 42157 42051 0.25%

4

Case #

4,

MBB4

2

60.0 18.0 135 50.8 0.90 0.10 1.92 43847 41684 5.19%

54.5 18.7 134 47.5 0.96 0.11 1.88 43534 41505 4.89%

63.1 18.2 116 48.8 0.91 0.27 2.50 42633 42435 0.47%

3

63.7 20.7 130 52.9 0.95 0.19 2.03 43433 42743 1.61%

64.7 18.0 116 52 0.90 0.10 2.28 43347 42485 2.03%

63.5 18.8 107 55.5 1.00 0.22 2.38 42974 42721 0.59%

4

66.1 18.0 120 50.5 0.90 0.10 1.85 43782 42941 1.96%

63.7 20.7 130 52.9 0.95 0.148 2.03 43706 43004 1.63%

68.8 19.2 81.7 54.1 1.03 0.17 2.17 43695 42301 3.30%


Table 8. Effect of applying the multi-burn base bleed for the studied cases

No. case

designation

No. of

Slots, Ns

Range of original

base bleed, X [m]

Range of multi-burn

base bleed, X [m]

X

[%]

1 Case # 1 Ns = 2 40516.2 41607 12% Ns = 3 41236.8 41818 6%

Ns = 4 41717.5 41915 2%

2 Case # 2

Ns = 2 40703 41657 10.4%

Ns = 3 41440 41857 4.2%

Ns = 4 41872 41960 0.9%

3 Case # 3

Ns = 2 40728 41687 10.4%

Ns = 3 41613 42053 4.4%

Ns = 4 42131 42154 0.2%

4 Case # 4

Ns = 2 41006 42435 15.0%

Ns = 3 41905 42743 8.1%

Ns = 4 42494 43004 4.6%

Table 9. Effect of number of slots for all studied cases

No. case

designation

No. of

Slots, Ns

Range of original base

bleed, X [m]

Range of multi-burn base

bleed, X [m]

Ns = 2 Ns = 3 Ns = 4 Ns = 2 Ns = 3 Ns = 4

1 Case # 1 Range

40516 41236.8 41717.5

41607 41818 41915

X [%] 8.0% 13.3% 2.1% 3.0%

2 Case # 2

Range 40703

41440 41872 41657

41857 41960

X [%] 8.0% 12.7% 2.0% 3.0%

3 Case # 3 Range

40728 41613 42131

41687 42053 42154

X [%] 9.6% 15.2% 3.6% 4.6%

4 Case # 4 Range

41006 41905 42494

42435 42743 43004

X [%] 9.5% 15.7% 2.8% 5.2%


Figure 8 represents the predicted time change of injection parameter, for the case MBB 2 with

different number of slots compared with the cases MBB 1, PBB 1 and PBB 2 when Ns = 2. In

all cases of MBB 2 injection parameter is higher than in case of the other cases in the first

seconds and less in the remaining time. For all cases of MBB 1 and MBB 2, injection

parameter takes the shape of 2 steps, the first one represents the burn time of the grain part

with BR2, while the second one represents the burn time of the grain with BR1 which starts

from the beginning. In general, injection parameter in case of MBB 1 and MBB 2 are much

closer to optimum value when compared with the injection parameter of the original grain.

Also, the application of multi-burn grain with the increase of Ns gets the pattern of injection

more flatten leading to increase in range. For MBB 1 the burn time is longer than the case of

MBB 2 as result of the lower BR1 (see table 7)

The benefit of base bleed on Mach number of the projectile when using

multi-burn for Ns = 2 is shown in Fig. 9.

Table 10. The effect of number of variables on range

No. Case

designation

Range increase with respect to case #1, X [%]

Ns = 2 Ns = 3 Ns = 4

PBB MBB PBB MBB PBB MBB

1 Case # 2 2.1% 0.5% 2.1% 0.4% 1.5% 0.4%

2 Case # 3 2.4% 0.8% 3.9% 2.3% 4.0% 2.3%

3 Case # 4 5.4% 8.2% 6.9% 9.0% 7.6% 10.5%


Fig. 8. Predicted time change of injection parameter, for the case MBB 2 with different

number of slots compared with the cases MBB 1, PBB 1 and PBB 2 when Ns = 2.

Fig. 9 Change of projectile Mach number versus time of flight at the first case of multi-

burn base bleed grain with 2 slots compared with the first case of the ordinary base bleed

grain with 2 slots.

Figure 10 shows the response surface of range with L% for different BR2 in case of Ns = 2

(a) and Ns = 4 (b) for the case MBB 2. For both values of Ns and different values of BR2,

range increases with the increase of L% for all values of BR2 up to optimum value. More than


this value the range decreases. The optimum value of L% increases with the decrease of BR2.

For the same values of L% and BR2 except for BR2 = 2.5 and L% > 0.4, the range is higher

in case of Ns = 4 when compared with the value in case of Ns = 2. In case of Ns = 4, it is

more favorable the lower limits of BR2. This may related to the incorporation of increasing

the number of slots.

Figure 11 shows the predicted time change of injection parameter for the fourth case of multi-

burn base bleed grain with different number of slots and compared with the original base

bleed grain. In all cases of MBB 4, the injection parameter is higher than in case of the

original base bleed grain for the first seconds and less for the remaining time. The benefit of

that is as the same as mentioned in Fig.5. For all cases of MBB 4 injection parameter takes the

shape of 2 steps. The first one represents the burn time of the grain part with BR2, while the

second one represents the burn time of the grain part with BR1 which starts from the

beginning. Also, burn time in all cases of MBB4 is longer than the time in case of original

base bleed grain. This is related to higher Rmax in the cases of MBB 4 and lower BR1 when

compared with the corresponding values of the original grain.

The benefit of base bleed on Mach number of the projectile when using

multi-burn for the case MBB 4 and Ns = 4 is shown in Figure 12.

Figure 13 shows the surface response of the predicted change of range with the Rmax for

different values of L in case of MBB 4 for Ns = 2 and Ns = 4. For all values of L, the range

increases with the increase of Rmax up to optimum value and after this value the range

decreases. The optimum value of Rmax decreases with the increase of L. For the same Rmax,

the range increases with the increase of L. The effect of outer radius change on range is higher

than the effect of length change.

Figure 14 shows the predicted change of range with BR1 in the case MBB 3, Dexit = 40 mm

and 48 mm when Ns = 2 for the lower, medium and upper limits of both BR2 and L%. It can

be seen that the range for all cases increases with the increase of BR1 up to certain value and

then it decreases. The value of BR1 that corresponds to maximum range decreases with the

increase of the limits of BR2 and L% for the same Dexit. However it is higher in case of Dexit =

48mm for all Limits cases. The range is higher for all BR1 with the increase of the limits of

both BR2 and L% from low to medium but it decreases when the limits go to the upper limits.

The justification of this could be as follows; in case of low limits, it requires higher value of

BR1 to get optimum injection parameter. For the

Dexit = 48 mm, the accumulation of the pressure inside the base bleed chamber is less than in

case of Dexit = 40 mm leading to reduction in mass flow rate [9] and consequently higher BR1


is required to get near to optimum injection parameter. With the increase in the limits from

low to medium, injection parameter get closer to the optimum value but when it reaches the

upper limits, injection parameter exceeds optimum value leading to reduction in base bleed

efficiency. Also, with the increase in the limits of both BR2 and L% more mass flow rate is

generated and larger Dexit is preferred [9].

Conclusion

It’s found that after optimization of all parameters the lower value of Rin is preferred.

Meanwhile, higher values of Dexit are required for longer range to allow the high mass

flow rate to stream with reasonable injection parameter.

The optimum value of BR1 is variable depending on the grain configuration and

number of slots. But the value decreases with the increase of Ns. Also it decreases with

the increase of the studied variables as another source of mass flow generation such as

Rmax, L, BR2 and L% rate is effective besides BR1.

The increase of number of slots increases the range for all studied cases of PBB,

MBB. However the percentage of range increases is less for MBB when compared

with the corresponding percentage of PBB

The application of multi-burn base bleed enhances the range regardless the number of

slots and the studied cases when compared with the corresponding optimized original

grain cases. The maximum increase of the range is found to be 15% in case MBB 4

with Ns = 2.

The increase of number of variables in the optimization process increases the

calculated range reaching the maximum value in case if MBB 4 with Ns = 2 and 4.

This reveals the benefit of optimization process.

When applying MBB, for different values of BR2, the range increases with the

increase of L% up to its optimum value. More than this optimum value the range

decreases. The optimum value of L% increases with the decrease of BR2. For the

same values of L% and BR2, the range is higher in case of Ns = 4 when compared

with the value in case of Ns = 2. In case of Ns = 4, it is more favorable than lower

limits of BR2.

For all cases, the range increases with the increase of Rmax up to optimum value and

beyond this value the range decreases. The optimum value decreases with the increase

of L. For the same Rmax, the range increases with the increase of L. The effect of outer

radius change on range is higher than the effect of length change.


(b) (a)

Fig. 10 The response surface of range with grain length ratio for different BR2 for the case MBB 2 when

Ns = 2 (a) and Ns = 4 (b)

Fig. 11. Predicted time change of injection parameter, for the case MBB 4 with different number of slots

compared with the case PBB 1 with Ns = 2

Grain length ratio,

L%


Fig. 12 Change of projectile Mach number versus time of flight at the fourth case of multi-burn base

bleed grain with 4 slots compared with the original base bleed grain.

(b) (a)

Fig. 13 The response surface of range with Rmax for different L in case of MBB4 for Ns = 2 (a)

and Ns = 4 (b)


Fig. 14. The predicted change of range with BR1 in the case MBB 3, Dexit = 40 mm and 48

mm when Ns = 2 for the lower, medium and upper limits of BR2 and L%

References [1] K. Anderson, N. E. Gunners and R. Hellgren, "Swedish Base Bleed - Increasing

the Range of Artillery Projectiles through Base Flow", Propellants and

Explosives I, 69-73 (1976).

[2] J. K. Fu and S. M. Liangt," Drag Reduction for Turbulent Flow over a Projectile:

Part I, "Journal of Spacecraft and Rockets, Vol. 31, No. 1, pp. 85-92 (1994).

[3] T. Mathur and J. C. Button, "Base-Bleed Experiments with a Cylindrical

Afterbody in Supersonic Flow", Journal of Spacecraft and Rockets, Vol. 33, No.

1, pp. 30-37 (1996).

[4] Y. K. Lee, H. D. Kim and S. Raghunathan, "Optimization of Mass Bleed for

Base-Drag Reduction", AIAA Journal, Vol. 45, No. 7, pp. 1472-1477 (2007).

[5] H. Bournot, E. Daniel and R. Cayzac, "Improvements of the base bleed Effect

Using Reactive Particles", International Journal of Thermal Sciences, Vol. 45,

pp. 1052–1065 (2006).

[6] A. Davenas, "Solid Rocket Propulsion Technology", Pergamon Press Ltd.,

England, pp. 350 (1993).

[7] Slobodan jaramaz and milojko injac, "Effect of grain characteristics on range of

artillary projectiles with base bleed", First International Symposium on Special

Topics in Chemical Propulsion, Athens, Greece, 23-25 November (1988).

[8] H. A. Abou-Elela, A. Z. Ibrahim, O. K. Mahmoud and O. E. Abdel-Hamid,

"Ballistic Analysis of a Projectile Provided with Base Bleed Unit", 15th

International Confrence on Aerospace Scienses & Aviation Technology,ASAT-

15, Egypt, 28 -30 May (2013).

[9] H. A. Abou-Elela, A. Z. Ibrahim, O. K. Mahmoud and O. E. Abdel-Hamid,

"Effect of Base Bleed Dimensions on the Ballistic Performance of Artillery

Projectile" 16th

International Confrence on Applied Mechanics and Mechanical

Engineering, AMME-16, Egypt, 27-29 May (2014).


[10] J. Danbergand, "Analysis of the Flight Performance of the 155 mm M864 Base

Burn Projectile", Technical Report BRL-TR-3083, US Army Ballistic Research

Laboratory (1990).

[11] K. Andersson, "Different Means to Reach Long Range > 65 Km, for Future

155mm Artillery Systems. Possibilities and Limitation", 17th Int. Sympo. on

Ballistics, Midrand, South Africa, 23-27 March (1998).

[12] A.Z. Ibrahim, "Ballistic Performance of Base Bleed Unit", M. Sc. Thesis, MTC

(2000).

[13] JMP, Design of Experiments, Release 6, SAS Institute Inc., Cary, NC, USA,

2005.

[14] Myers R. H., Montgomery D. C., "Response Surface Methodology

– Process and Product Optimization Using Designed Experiments", Wiley Series

in Probability and Statistics, New York, 1995.

[15] Nuran B., Yi C. “The Response Surface Methodology ", M.Sc. Thesis,

University of South Bend, USA, 2007.

[16] Conn A.R., Scheinberg K., Vicente L.N., Introduction to Derivation-Free

Optimization, Society for Industrial and Applied Mathematics and the

Mathematical Programming Society, 2009.

[17] Kaymaz I., McMahon C. A.,"A Response Surface Method Based on Weighted

Regression for Structural Reliability Analysis" Journal of Probabilistic

Engineering Mechanics Vol. 20, pp. 11–17, 2005.

[18] Jasbir S. A., Introduction to Optimum Design, Elsevier Academic Press, second

edition, 2004.

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